## Approachability at the second successor of a singular cardinal.(English)Zbl 1184.03046

Summary: We prove that if $$\mu$$ is a regular cardinal and $$\mathbb P$$ is a $$\mu$$-centered forcing poset, then $$\mathbb P$$ forces that $$(I[\mu ^{++}])^{V}$$ generates $$I[\mu ^{++}]$$ modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

### MSC:

 300000 Other combinatorial set theory 3e+35 Consistency and independence results

### Keywords:

approachability ideal; internally approachable
Full Text:

### References:

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